Question

If G is a group ab $\text{∈}$Z (G) , prove that ab=ba.

Short answer

Answer:

It is proved that if $ab\in Z\left(G\right)$, then ab=ba.

Step-by-step solution

Step by Step Solution: Step 1: Center of group

We have thedefinition of center of a groupthat:

Let G be a group. Then, the center of the group G is denoted by Z(G) and defined by,

$Z\left(G\right)=\left\{\left.a\in G/ag=ga,\forall g\in G\right\}\right.$

It means that an element of is in if and only if it commutes with every element of .

Show that ab=ba

Let$a,b\in G$, then:

$\begin{array}{c}ab\in Z\left(G\right)\\ ba\in Z\left(G\right)\dots \left(\text{From\hspace{0.17em}\hspace{0.17em}definition\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}}Z\left(G\right)\right)\\ ab=ba\dots \left(\because \text{\hspace{0.17em}From\hspace{0.17em}\hspace{0.17em}the\hspace{0.17em}\hspace{0.17em}definition\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}}Z\left(G\right)\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}elements\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}}Z\left(G\right)\text{\hspace{0.17em}\hspace{0.17em}are\hspace{0.17em}\hspace{0.17em}commutative}\right)\\ aab=aba\\ ab=ba\dots \left(\text{By\hspace{0.17em}\hspace{0.17em}left\hspace{0.17em}\hspace{0.17em}cancellation\hspace{0.17em}\hspace{0.17em}law}\right)\end{array}$

Hence, it is proved that ab=ba.